Codes and

Reed-Solomon

Codes and

Their Applications

Edited

by

Stephen

B.

Wicker

Georgia

InsEitute

of Technology

Vijay K.

Bharga va

University

of Victoria

.

IEEE

\/

PRESS

IEEE Communications

Society

and IEEE Information

Theory

Society,

Co-sponsors The

Institute

of Electrical

and Electronics

Engineers,

Inc., New York

Irwinn

S.

Reed

and Gustave

Solomon

t iflUst

four

SUCCCsjC

collisions

can

occur between

two

patterns,

Which alleviates

the

burden

onthe

error control

system

as

vell.

Further

details

ofthis

construction

canbe

found

in[2].

2.10 Vajda’s

Construction

Theuseofa

product

code construction

of

hopping

patterns

hasalso

been explored

by

Vajda

[32]

who took

a

cyclic product

oftwo

codes.

Thefirst

code

is

obtained

from C(N,

t+1;0)

over GF(q),

where

Nisa

prime

As

described

in

Section

2.5,q’

hopping

patterns

of

period

Ncanbe

obtained from

this

code.

Letr

denote

the

largest integer such

that

N’

<qt

This code

isa

(nonlinear)

cyclic code consisting

ofasetof

N1 hopping

patterns

of

period

Nandall

their cyclic shifts.

Note

thatthe

minimum

distance

ofthis

nonlinear

cyclic code

isN—t.The

other code

isa

coset

of

C(M,

K1;2)

over GF(Nr)

thatisa

subcode

of

C(M, K+1;0)

over GF(Nr).

Note

thatMisa

divisor

ofN

—1.

Thus,

this

code

has

Nr(K_1)

code words over

an

alphabet

ofsizeiVr,and

since

the

code words belong

to

C(M, K+1;0),the minimum

distance

between

cyclic

shifts

oftwo

code words

isat

least

M—K.

Vajda

has

proposed

using

the

cyclic product

of

these codes instead

ofthe

direct product

discussed

inthe

previous

subsection.

Thus, each

oftheM

NrvaIued

symbols

ina

code word

ofthe

second

code

is

replaced

bya

column

vector

of

length

N

consisting

ofa

code word

ofthefirst

code.

This creates

anNxM

matrix

Q

Q1.

Now,

MandNare

relatively

prime,

and

hence

the

entries

in

Q

canbe

read off

in

cyclic fashion

to

form ahopping pattern

of

length

MN

whose

ith

symbol

is

Q

mod

N,I

mod

M.

Since

the

cyclic product

ofan(Ill,k1,d1)

cyclic code with

an

(112,

k2,d2)

cyclic code

isan

(Iifl2,

k1k2, d1d2) cyclic code [13],

this

hopping

pattern

is

actually

a

code word

inan

[MN,

(k+

1)(K +I),(M—

K)(N

—t)]

cyclic code over GF(q).

There

are

Nr(K_1)

such hopping

patterns,

andit

follows

from

(8)

that,

as

shown

in

[32], Hmax<MN_(M_K)(Nt)AIK

Asan

example

ofthis

construction,

letq=32,N=31,t=2,r=3,andM

(3J3

—

1)/(31

—1)

993.

LetK4.

Then, aset

of

3132

=

887.

503.

681

hopping

patterns

of

period

31

‘993

30,783

over GF(32)

is

obtained.

The

maximum

Hamming

correlation

is

2102,

sothat

there

is,onthe

average,

onehit

every 14.64 symbols.

In

contrast,

the

Reed

and

Solomon

setsof

hopping

patterns

from C(31,

3;0)

over GF(32)

provide

1024 hopping

patterns

of

period

31 with amaximum

Hamming

correlation

of2,thatis,onehit

eery

15.5 symbols,

which

is

very slightly

better.

2.11 Einarsson’s

Construction Because

of

technological

limitations

onthe

frequency

synthesizers

used

to

produce

the

frequency-hopped

signals,

the

hopping

rateina

FuSS

system

is

limited toafew

thousand

dwells

per

second.

Inafast

FHJSS system,

the

transmission

ofa

symbol

occurs over several

hops, anditis

necessary

touse

ji.-ary signaling

in

order

to

achieve

areasonably

large

data

rate.

Einarsson

j5]

proposed

acombined

design

of

hopping

patterns

and

M-ary modulation

forusein

such systems.

In

systems

using

this

design,

each transmitter

is

assigned

acollection

ofM

hopping

patterns

of

length Nand transmits

one

M-ary

data

symbol

perN dwells

by

choosing

and

transmitting

oneofthe hopping

patterns.

Thedatarateis

thus log2(M)/NT,,

bitsper

second.

Note, however,

thatthe

receiver

is

now more complicated

since

it

must track

allM

possible

hopping

patterns

in

order

to

determine

which

oneis

being transmitted.

Thus,

M

different

frequency

synthesizers

might

be

needed

in

each receiver.

The

Einarsson

design

uses allthe

nonzero

code words

inthe

Reed- Solomon

code C(q—1,2;0).

Each transmitter

is

assigned

allthe code words ina cyclic equivalence

class.

Thus, M=N=q—1,andthe

hopping

patterns assigned

tojth transmitter

areofthe

form

(j,j

j)+aL(1,a,a_

q—2)

0j

sq—

1.

Since

all

these sequences

are

from C(q—1,2;0),the

number

ofhits between

two

patterns

assigned

to

different

transmitters

isat

most 1regardless

ofthe relative

time delay between

thetwo

patterns.

However,

the

number

ofhits

per

period

canbe

guaranteed

tobe1

only ifthetwo

transmitters

are

frame synchronous.

Ifthe

transmitters

are

only dwell-synchronous,

then thetailend andthe front endofIwo

possibly

different

hopping

patterns

from

an

interfering transmitter

can

cause collisions,3

and

thus

the

number

ofhitsper

period

can

betwoin

some cases.

There isalsothe

question

ofthe

initial acquisition

of

synchronization

inthe

receivers

in

such systems

since

the

hopping

patterns assigned

toa

transmitter

arenot

cyclically

inequivalent.

In

fact,

the

Ham ming cross-correlation

between

two

hopping

patterns

assigned

tothe

same transmitter

can

have values

as

large asN—1.

2.12 Other Constructions There

are

several

other constructions

of

frequency

hopping

patterns

that

arenot

directly

related

to

Reed-Solomon

codes except

in

certain special cases.

3A

similar

phenomenon

in

DS/SS

systems

gives ri5etotheodd

cross-con-eladon

function

of

binary sequences

(cf.

(231).